In the US, your copy of the marriage certificate is probably certified. Also, with a Spouse visa, you will quickly learn to compile a 'Spouse renewal' file folder. Makes annual renewal of husbands extension 100 times easier for the Thai wife. Keeps all the documents, copies, pics, hand drawn maps, and related stuff in one place.
Depends on your home Province and district. May require an official Thai translation (meaning notarized); the folks who do the official translation can provide the notary. Then register your marriage in your home district office; will need to provide current updates of active marriage from local district for O-spouse renewal of extension.
When I was in your situation in 2006, we got a US fiancee visa. Took several months, but worked out in the long run. Then, marriage in US; then a green card, followed by citizenship. Upon return to Thailand, registered our marriage in/with Thailand.
I use Google Voice; works from anywhere in the world to call a USA number. Not sure about calling other places . . cause I never call other places. Use a Thai phone here in Thailand, but have wifey make the calls. BTW, Google Voice phone setup is $0 cost. Been using it for 10+ years.